Sharp and unsharp observables on σ-MV algebras—A comparison with the Hilbert space approach

We prove that any observable with a standard value space on a probability MV algebra M can be extended from a mapping defined on measurable subsets with values in M to a mapping defined on bounded real valued measurable functions with values in the set of sharp observables associated with M. Our method of proof is quite different from the method of the proof of analogous result for POV-observables on a Hilbert space. We show that for POV-observables with commuting range both approaches give the same results.